T. J. Rivlin

Chebyshev polynomials : from approximation theory to algebra and number theory

Definitions and Some Elementary Properties Extremal Properties Expansion of Functions in Series of Chebyshev Polynomials Iterative Properties and Some Remarks About the Graphs of the Tn Some Algebraic and Number Theoretic Properties of the Chebyshev Polynomials References Glossary of Symbols Index.

A Survey of Optimal Recovery

The problem of optimal recovery is that of approximating as effectively as possible a given map of any function known to belong to a certain class from limited, and possibly error-contaminated, information about it. In this selective survey we describe some general results and give many examples of optimal recovery.

The optimal recovery of smooth functions

SummaryIt is shown that there is a positive lower bound,c, to the uniform error in any scheme designed to recover all functions of a certain smoothness from their values at a fixed finite set of points. This lower bound is essentially attained by interpolation at the points by splines with canonical knots. Estimates ofc are also given.

Ergodic and mixing properties of Chebyshev polynomials

The Chebyshev polynomial of degree n is defined as T„(x) —cosnd, where x = cos 6, O^d^w. If we put T0(x) = l/\/2 then it is well known that the sequence {Tk(x): fe = 0, 1, 2, • • • } is a complete orthonormal set (c.o.n.s.) in L2(X, 93, p). Let Tn be the transformation defined by T„(x), then Tn is a measurable almost everywhere n to one mapping of X onto itself satisfying Tn(Tm) = Tnm for n, m = 1, 2, • • • , i.e. {P„:w=l, 2, • ■ • } is a semi-group under composition. (The semi-group property is a simple consequence of the definition of the Chebyshev polynomials.)

The growth of polynomials bounded at equally spaced points

If the absolute value of a (real) polynomial of degree d is bounded by 1 at k equally spaced points of the real line, it is of interest to know how large its absolute value on the interval spanned by the points can be. This work provides a fairly definitive answer to this question.

Some numerical experiments in the theory of polynomial interpolation

An important unsolved problem in the theory of polynomial interpolation is that of finding a set of nodes which is optimal in the sense that it leads to minimal Lebesgue constants. In this paper results connected to this problem are obtained, and some conjectures are presented based upon numerical evidence garnered from extensive computations.

Approximation by circles

The problem considered is to assign a measure of circularity to a given compact set in the plane. The measure adopted is the size of the smallest annulus containing the given set. Two different notions of the size of an annulus, that of area and that of difference of radii are studied.ZusammenfassungDas hier untersuchte Problem ist, einer kompakten Menge in der Ebene ein Maß der Kreisförmigkeit zuzuschreiben. Als Maß wird die Größe des kleinsten Kreisringes gewählt, der die gegebene Menge enthält. Zwei verschiedene Größenbegriffe für den Kreisring werden untersucht, nämlich dessen Oberfläche und die Differenz der Radien.

Approximation of monomials by lower degree polynomials

Our topic is the uniform approximation ofxk by polynomials of degreen (n<k) on the interval [−1, 1]. Our major result indicates that good approximation is possible whenk is much smaller thann2 and not possible otherwise. Indeed, we show that the approximation error is of the exact order of magnitude of a quantity,pk,n, which can be identified with a certain probability. The numberpk,n is in fact the probability that when a (fair) coin is tossedk times the magnitude of the difference between the number of heads and the number of tails exceedsn.

Theoretical calculations in gaseous detonation

Abstract Nistenxier Chapman Jougurt, strong and weak detonation properties have been obtained for a five atom [C. H. O. N. A.] system. A digital computer programme has been developed which determines detonation properties in steady one-dimensional equilibrium flow. The equations and method of solution are set forth. Equilibrium and frozen sound speeds are calculated for the hot product mixtures. The effect of initial conditions and comparisons with experiment are given.

Optimally Stable Lagrangian Numerical Differentiation

Approximating a derivative of a function at a point by the corresponding derivative of its interpolating polynomial is called Lagrangian numerical differentiation. In this work we describe the nodes at which to interpolate in order to minimize the error in Lagrangian numerical differentiation due to errors in the sampled values of the function.